Bone on one side distorts the other side of the mesh
This issue likely is occurring because vertices on one side have been assigned to bones on the opposite side.

When using a mirror modifier, this issue can be easily rectified by simply un-assigning all vertices from the right side—this can be done manually through the Vertex Groups area in the object data panel (just select all vertices in edit mode and click "remove" for all bone-related vertex groups whose name ends in ".r")

Another option (when using the mirror modifier) is to go through each of the right-side bones and subtract the weight from the left side. (However, when using a mirror modifier this will NOT work by selecting the left-side bones and removing weight from the right.)

If the mirror modifier is not in use, just do the second process on both sides—i.e. remove from the right bones vertices on the left, and remove from the left bones vertices on the right.